TITLE: Morita equivalence base equivalence relation for discrete groups
QUESTION [7 upvotes]: In the context of "discrete groups", is there an equivalence relation that implies the Morita equivalence of their reduced group C*-algebras?
We define $G \sim H$ for discrete groups $G$ and $H$, when $C^*_r (G)$ and $C^*_r (H)$ (or even their full group C*-algebras) are Morita equivalent as the two C*-algebras.
Does $\sim$ have an explicit description?
thanks
PS: By Alain Valette's first comment in below, the $C^*_r (-)$ case shouldn't be considered so the problem is about $C^*(-)$.

REPLY [6 votes]: Not really an answer, but too long for a comment: let $G,H$ be countable, abelian, torsion-free groups. Claim: $C^*G\sim C^*H$ if and only if $G$ is isomorphic to $H$. To prove the non-trivial implication: denote by $\hat{G}$ the Pontryagin dual of $G$, a compact connected abelian group. Then $C^*G\simeq C(\hat{G})$ by Fourier transform. Since $C^*G\sim C^*H$, the groups $\hat{G}$ and $\hat{H}$ are homeomorphic, hence the first $\check{C}$ech cohomology groups $H^1(\hat{G},\mathbb{Z})$ and $H^1(\hat{H},\mathbb{Z})$ are the same. It remains to see that $H^1(\hat{G},\mathbb{Z})\simeq G$. Write $G=ind-lim_k G_k$, an inductive limit of finitely generated free abelian groups, i.e $G_k\simeq \mathbb{Z}^{n_k}$. Dualizing, $\hat{G}$ is a projective limit of tori, i.e. $\hat{G}=proj-lim_k \mathbb{T}^{n_k}$.Since $\check{C}$ech cohomology converts projective limits into inductive limits, we get $H^1(\hat{G},\mathbb{Z})=ind-lim_k H^1(\mathbb{T}^{n_k},\mathbb{Z})=ind-lim_k \mathbb{Z}^{n_k}=G$.